Question

Taking a test i don’t know anything about please help

In an Economics course, the correlation between the students' total score prior to the final exam and
their final exam score is r = 0.60. The pre-exam totals for all students in the course have mean 280
points and standard deviation 30 points. The final exam scores have mean 75 points and standard
deviation 8 points.
4. Find the equation of the least-squares regression line for predicting a final exam score from a
student's pre-exam total.
5.
Interpret the slope in context.
6.
Interpret the y-intercept in context.
7.
Calculate the fraction of variability, R?.
8.
Interpret R? in context.

Answer 1

See below for answers and explanations

Explanation:

Problem 4:

The line of regression is
`\hat y = a+bx` where:

`a=\overline y-b \overline x`

`b=(r*s_y)/(s_x)`

We are given that
`\overline y=75`,
`s_y=8`,
`\overline x=280`,
`s_x=30`, and
`r=0.60`, therefore our slope,
`b`, is:

`b=(r*s_y)/(s_x)`

`b=((0.60)(8))/(30)`

`b=0.16`

Therefore, the slope of the regression line is 0.16, which can be used along with the values of
`\overline y` and
`\overline x` to find the constant
`a`:

`a=\overline y-b \overline x`

`a=75-(0.16)(280)`

`a=30.2`

This means our final regression line is
`\hat y = 30.2 + 0.16x`

Problem 5:

The slope,
`b=0.16`, means that for every 1 point earned for a student's pre-exam total, their final exam score will increase by 0.16 points for each point they earned on the pre-exam.

Problem 6:

The y-intercept (or constant),
`a=30.2`, means that if a student's pre-exam total were 0, then they would expect to get a 30.2 on the final exam.

Problem 7:

`R^(2)=(0.60)^(2)=0.36=36\%`

Problem 8:

The fraction of variability (aka. coefficient of determination),
`R^2`, means that a certain proportion (or percentage) of the variance in the response variable can be explained by the explanatory variable. In context, this means that 36% of the variance in final exam scores can be explained by the pre-exam scores.